Given a velocity field u(x, t), we consider the evolution of a passive tracer \(\theta \) governed by \(\partial _t\theta + u\cdot \nabla \theta = \Delta \theta + g\) with time-independent source g(x). When \(\Vert u\Vert \) is small in some sense, Batchelor, Howells and Townsend (J Fluid Mech 5:134, 1959) predicted that the tracer spectrum scales as \(|\theta _k|^2\propto |k|^{-4}|u_k|^2\). In this paper we prove that, for random synthetic two-dimensional incompressible velocity fields u(x, t) with given energy spectra, this scaling does indeed hold probabilistically, asymptotically almost surely for large |k| and small \(\Vert u\Vert \). We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of u.

中文翻译：

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示踪剂湍流：重访Batchelor–Howells–Townsend频谱

给定一个速度场ù（X， 吨）中，我们考虑一个被动示踪剂的进化\（\ THETA \）管辖\（\局部_t \ THETA + U \ CDOT \ nabla \ THETA = \德尔塔\ THETA +克\ ）和与时间无关的来源g（x）。当\（\ Vert u \ Vert \）在某种意义上较小时，Batchelor，Howells和Townsend（J Fluid Mech 5：134，1959）预测示踪剂光谱的缩放比例为\（| \ theta _k | ^ 2 \ propto | k | ^ {-4} | u_k | ^ 2 \）。本文证明，对于随机合成的二维不可压缩速度场u（x，  t）在给定能谱的情况下，对于大| k | 和小\（\ Vert u \ Vert \）。我们还提出了由u的时间依赖性引起的BHT标度的渐近校正因子。